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Distribution and normality of the decimal digits of π

Determine the statistical distribution of the decimal digits of π; in particular, establish whether π is normal in base 10, meaning that each digit and each finite block of digits occurs with limiting frequency 10^{-k}.

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Background

The paper uses the Sevcik fractal dimension estimator to analyze long prefixes of the decimal expansion of π and compares them to uniformly distributed random sequences. To conduct statistical comparisons, the authors explicitly note that the distribution of π’s digits is not known and therefore employ inequalities that do not assume normality.

The question of whether π is normal in base 10 is a classical open problem in number theory; resolving it would determine whether the digits of π are uniformly distributed and justify certain statistical modeling assumptions used in empirical analyses like those discussed in the paper.

References

Since we do not know the distribution of the decimals of $ \pi $, the approximation of the inequality of Vysochanskij–Petunin was used to compare the diverse values of $ D_S $ obtained when the decimals of $ \pi $ studied were increased by taking the first in the sequence of $ 10, 100, 1000, \ldots, 107, 108, 109 $ $\pi$ digits and a sequence of the same length of decimal distributed uniformly as $ U_Z(0.9) $ or $ U_R(0.1)) $ versus the randomized sequence of the same number of $ \pi $ digits.

— Fractal dimension, and the problems traps of its estimation (2406.19885 - Sevcik, 27 Jun 2024) in Subsubsection: Sevcik’s fractal dimension convergence to D_{HB} (discussion of π digits)